What is Energy?

Brokk Toggerson

11 What is Energy?

Brokk Toggerson

Learning Objectives

By the end of this chapter, you should be able to…

Define work.
Define a force.
List the SI unit of force.
List some touchstone values of force.
Define pressure.
List some units of pressure.
Define mass.
Compare mass and weight.
List the SI unit of mass.
List some touchstone units of mass.
List units of energy.
List some touchstone values of energy.

What is energy? If you look in most textbooks (biology, chemistry, or physics!), you will see the following definition.

We define energy as the ability to do work.

–OpenStax Biology 2e, Chapter 6.2

This definition doesn’t really answer our question however, it just kicks the can down the road. Now we are left to ask what is work? The aforementioned OpenStax Biology 2e, does not actually define this quantity! If we, instead, look in a physics text, we find the following definition.

Work is the transfer of energy by a force that causes an object to move a distance; the product of the component of the force in the direction of the distance traveled displacement and the magnitude of the distance.

–OpenStax College Physics 2e. Chapter 7 glossary

Once again, we have just kicked the can down the road! We probably have a good concept of what distance is, but what actually is a force? We may have some intuitive definition, but our experience with entropy gives us reason to doubt the accuracy of our intuitive definitions; before studying entropy, you may have thought that entropy was “disorder,” a definition we have found to be false. Is the same true with our intuitive definition of force?

What is a Force?

This may be one of the most important sections in this entire book! Make sure you read and understand it carefully!

Put in its most simple terms, a is a push or a pull. When you push on an object, you exert a force on it. When someone pulls on your arm, they are exerting a force on you. When you jump, you push off the ground, you exert a force on it.

Clearly forces have both a size (magnitude) and a direction: a tiny flick is different from a hard punch, and pushing to the left on an object yields a different behavior than pushing to the right.

Key Takeaways

A force is a push or a pull, and has both a magnitude (size) and direction.

Make sure you keep this in mind throughout this course!

Units of Force

The unit of force in the SI system is the Newton ( $\rm N$ ) named after Sir Issac Newton (much more on him later in the course!). In the Imperial system of measurement used in the U.S. and a (very) few other countries, the unit of force is the pound ( $\rm lb$ ).

$0.225 \, \mathrm{lb} = 1 \, \mathrm{N}$

Touchstone Amounts of Force

You probably already have a feel for units like meters, but for the new units like Newtons, it is often useful to have a few touchstone values to help give you a sense of size. We will be providing these touchstone examples throughout this course when we encounter new units (we didn’t have to do it for the previous unit as entropy is fundamentally just the number of combinations!). You are expected to memorize these touchstone values and they will be on your quizzes! Below, you can see a table of a few such touchstone values of force. To interpret these tables, you will need to have your SI prefixes memorized as detailed in the “what do we expect you to know” section.

A few touchstone amounts of force from Orders of magnitude (force) – Wikipedia.
Item	Typical Forces [N]	Typical Forces [lbs]
Force needed to stretch DNA to 1.5x its initial length.	$10^{-13} \, \mathrm{N}$	$10^{-14} \, \mathrm{lbs}$
Forces of molecular motors.	$10^{-12} \, \mathrm{N}$	$10^{-13} \, \mathrm{lbs}$
Force needed to break molecular bonds.	$\sim \mathrm{nN}$	$10^{-10} \, \mathrm{lbs}$
Force with which a person standing still pushes on the ground.	$500 - 1000 \, \mathrm{N}$	$100 - 224 \, \mathrm{lbs}$
Peak force of a small gasoline powered car.	$\sim \mathrm{kN}$	$\sim 200 \, \mathrm{lbs}$
Peak force of a human weight lifter.	$\sim \mathrm{kN}$	$\sim 200 \, \mathrm{lbs}$

Another Useful Way to Think of Force: Pressure

Forces are well and good, but for liquids or gases, pressures are often more useful. Or, you might also think about the osmotic pressure of a fluid inside of a cell. Fundamentally, pressure is how much a force is spread over an area, meaning that it would have units of $\rm N/m^2$ . Some other common units of pressure that you might have seen are Pascals, atmospheres or mmHg, which are also described below. I can increase the pressure in two ways: more force or by putting that force over a smaller area. Pressure is why you can stand on a flat floor with no problems, but cannot step on a single nail without puncturing your foot: your skin has a threshold pressure beyond which it will puncture.

Example: Air Pressure

Problem:

As discussed below, air pressure is $101.3 \, \mathrm{kN/m^2}$ or $14 \, \mathrm{lbs/in^2}$ . If the surface area of a person is about $2 \, \mathrm{m^2}$ ^[1], then how much force does the air press on a person?

Solution:

SI System:

If the human body is about $2 \, \mathrm{m^2}$ and the pressure is $101.3 \, \mathrm{kN/m^2}$ then the total force is

$\left( 101.3 \frac{\rm kN}{\rm m^2} \right) \times \left( 2 \, \mathrm{m^2} \right) = 202.6 \, \mathrm{kN}$

Imperial System:

First we need to convert the human body area from $\rm m^2$ to $\rm in^2$ (if you are unsure how to do this, see Appendix F):

$\left( 2 \, \mathrm{m^2} \right) \times \left( \frac{100^2 \, \mathrm{cm^2}}{1 \, \mathrm{m^2}} \right) \times \left( \frac{1^2 \, \mathrm{in^2}}{2.54^2 \, \mathrm{cm^2}} \right) = 3100 \, \mathrm{in^2}$

Now we can compute the total force:

$\left( 14 \, \frac{\rm lbs}{\rm in^2} \right) \times \left( 3100 \, \mathrm{in^2} \right) = 43,400 \, \mathrm{lbs}$

Units of Pressure

Pressure is one of the most annoying quantities in science due to the large number of units in use. The standard unit associated with the SI system is the $\rm N / m^2$ which is also called the Pascal ( $\rm Pa$ ). However you will also see the following units used:

Units commonly seen for pressure and their conversions to the SI and Imperial systems. You do NOT need to memorize these values.
Unit	Conversion to $\rm N/m^2$ or $\rm Pa$	Conversion to the Imperial $\rm lbs/in^2$ ( $\rm psi$ )
Imperial Pounds per square inch (psi / $\rm lbs/in^2$ )	$6895$	–
Millimeters of Mercury (mmHg)	$133.3$	$0.19$
Atmospheres ( $\rm atm$ )	$1.013 \times 10^5$	14

The Atmosphere as a Unit of Pressure

The atmosphere is an important unit of pressure of particularly common use in the biological sciences. This is the the pressure pushing down on you from the weight of all of the air above your head. As you can see in the table above, $1 \, \mathrm{atm} = 101.3 \, \mathrm{kPa} = 14 \, \mathrm{psi}$ . This is a lot of force! Why doesn’t it crush you? Well your insides are also at atmospheric pressure, keeping you in equilibrium. What about SCUBA divers who go deep under water where the pressure is much higher? Well, they are breathing air at a higher pressure which keeps them closer to balance with the outside water. If, on the other hand, they were breathing air at simple atmospheric pressure, then bad things would happen as visible in the video below.

Example: Air Pressure on a Mountain

Problem:

Would you expect air pressure to be higher or lower at the top of a tall mountain such as Denali in Alaska?

Solution:

Air pressure arises from the weight of the air above you. At the top of a mountain, you are above some of the air so there is less air above you pushing down. Thus, we expect the air pressure at the top of a mountain to be less.

Mass (As Opposed to the Force of Weight)

You probably have some insight into mass: mass is the amount of stuff you have. One atom of gold has more mass than one atom of sodium because it has more protons, neutrons, and electrons. While this is similar to the everyday concept of weight, they are slightly different. As we will see in more detail in our unit on forces, is the force with which a planet pulls on an object. This is even visible in the units. The SI unit of mass is the kilogram while the SI unit of weight (being a force) is the Newton. In the imperial system, the unit of mass is the slug while the unit of force, as seen above, is the pound (lb).

Example: Converting from Pounds to Kilograms

Problem:

“One kilogram is about 2.2 pounds.”

-OpenStax Chemistry Atoms First 2e^[2].

Is this a true statement in the most pedantic sense of the word?

Solution:

No! If we are being as pedantic as possible, we cannot convert from pounds to kilograms! A kilogram is a unit of while a pound is a unit of . Thus, equating these two is like saying the moon is 1 second away from Earth. In both cases I have compared quantities with different units: force/mass in the kg/lbs example and distance/time with the Moon example.

With the addition of more information, however, I can compare the two. If we know that the one kg and is on Earth (usually true!) then we know the strength of the gravity on the object, can determine its weight (which is now a force!), and can then expressed that value in pounds. Similarly, if I know how fast I am traveling, then I can express a distance as a time. In our example of the Moon being 1 second away, this is true, if and only if, I am traveling at the speed of light $3 \times 10^8 \, \mathrm{m/s}$ . In both cases, more information is needed: what planet am I on, or how fast am I going.

Put another way, one kg would not have a weight of 2.2 lbs on a different planet. On the Moon for example, the weight of 1kg is only about 3.7 lbs.

This is not to put down the chemistry textbook or its authors! The assumption that you are standing on Earth is true for all but about 5 people at any given time. In other words, only about $6 \times 10^{-8} \%$ of people need to worry about this difference!

Example: A pound of bricks and a pound of feathers

Problem:

Consider the common brain teaser, “Which weighs more, a pound of feathers or a pound of bricks?” What is the answer? Which will have more mass?

Solution:

The two piles will always have the same weight by definition: “a pound is the same the world round.” In fact, it is even more generalizable than that! Being a unit of weight, a pound is, by definition, the same everywhere in the Universe. Thus, a pound of bricks on the Moon (where the pull of gravity is only 1/6^th that of Earth) still weighs the same as a pound of bricks on Earth.

What about the mass? If both piles are on the Earth, having the same weight will require that they have the same mass as they are both being pulled down by the same gravitational force (that of the Earth). However, this is NOT always true! Return to the case where the bricks are on the Moon and the feathers on on Earth. In this case, the two will NOT have the same mass! The pile of bricks on the Moon will need to be much bigger than it was on Earth to have the same pull down!

Touchstone Units of Mass (for those raised in the US!)

If you were raised in the U.S. you probably lack a “gut feeling” for kilograms. I know that in my experience, the kilogram was the hardest SI unit to “get a feel for” as the others have close imperial values which can be used for estimation:

One meter is about 1 yard. Technically a meter is just over 40in compared to the 36in for a yard, but this difference is usually negligible for everyday estimations.
One liter is about 1 quart. While a liter is not “technically” an SI unit, it is commonly used with the SI system.

These reference points are great for distance and volume. For mass, however, we have that annoying 2.2 conversion factor between pounds and kilograms described above (noting the caveats mentioned in the prior section). Thus, I provide the following table of touchstone masses. I expect you to have these memorized (they will be on your quiz!) as they will help you estimate the validity of your answers throughout this course.

A few touchstone amounts of force from Orders of magnitude (mass) – Wikipedia.
Item	Mass
Proton	$\sim 10^{-27} \, \mathrm{kg}$
Lead atom or Caffeine molecule	$\sim 10^{-25} \, \mathrm{kg}$
Average Human Cell	$\sim 1 \, \mathrm{ng}$
Grain of salt or sand, Mosquito	$\sim 1 \, \mathrm{mg}$
U.S. Dollar Bill	$\sim 1 \, \mathrm{g}$
Apple/Orange	$\sim 200 \, \mathrm{g}$
Laptop	$\sim 1 \, \mathrm{kg}$
Dr. Toggerson	$\sim 70 \, \mathrm{kg}$

Work and Energy

Now that we have clarified a force, we can return to the definition of : work is a force applied for a distance. Using our new definition of force from above, we can now clarify this: when I push or pull on an object for some distance, I am doing work on it, is then the capacity of an object to do work. While we will explore these ideas more fully in future chapters, we can already learn something from this definition: the units of work and energy.

Units of Work and Energy

Since energy is simply the capacity to do work, they must have the same units: whatever the units of work, they will be the same as the units of energy. Our definition above tells us that work is a force (push or pull) applied for some distance: a force and a distance. From our discussion of And and Or in Probability, we know that, at least in the probability context, “and” implies multiplication. This statement is, in fact, generally true: and usually translates to multiplication. Thus, the units of work, a force and a distance, should be the units of force times the units of distance: a newton times a meter $\rm N \cdot m$ . This unit is also called the Joule ( $\rm J$ ). While the Joule is the official SI unit, there are other units of energy in common use that you should know listed below. The conversion factors for these units will be on the data sheet you get for exams and thus you do not need to memorize them. We will mostly use Joules and electron-Volts in this course.

Unit	Commonly Used For	Conversion Factor
electron-Volt ( $\rm eV$ )	Atomic energy levels, molecular bonds, nuclear energies.	$1.602 \times 10^{-19} \, \mathrm{J} = 1 \, \mathrm{eV}$
food Calorie ( $\rm Cal$ )	Nutritional information	$1 \, \mathrm{Cal} = 4.184 \, \mathrm{kJ}$

Touchstone Amounts of Energy

Just like with Newtons above, knowing a few touchstone values for energy is also useful. You will notice that some of the values I want you to know are in $\rm eV$ while others are in Joules. While we do not expect you to know the $\mathrm{J} \leftrightarrow \mathrm{eV}$ conversion factor (see above), we do expect you to have these touchstone values memorized in whichever unit they are presented.

A few touchstone amounts of energy from Orders of magnitude (energy) – Wikipedia
Item	Typical amount of energy
Covalent bonds	$\sim 1 \, \mathrm{eV}$
Energy holding the nucleus together	$\sim 1 \, \mathrm{MeV}$
The energy of a small apple after falling 1m	$\sim 1 \, \mathrm{J}$
A punch	$\sim 10 \, \mathrm{J}$
The energy of a jumping person or the energy of a defibrillator shock	$\sim 100 \, \mathrm{J}$
Energy released from metabolizing 1g of food (carbohydrates, fat, or protein)	$\sim 10 \, \mathrm{kJ}$
Energy of a car moving at 90 kph (55 mph)	$\sim 100 \, \mathrm{kJ}$

A Note on Units

This is really the first topic in our course where units are an issue; entropy is the natural logarithm of a number of microstates. As such, the units are not super important. Yes, there is a $k_B$ with units $\rm J/K$ but that is purely a convention. There are some disciplines where the $k_B$ is dropped and entropy is simply a unit-less number.

Now that we are dealing with units, lets clarify a few things. We expect that you are familiar with converting units. You can check yourself using the related homework problems. If you need a bit of review, there is an appendix which covers these things. We also expect that you are familiar with the SI prefixes. Moreover, as educated science students, we will expect you to have memorized the SI prefixes from nano- to Giga- listed in the table below.

Prefix name (symbol)	Power of 10
nano- (n)	10^-9
micro- (μ)	10^-6
milli- (m)	10^-3
centi- (c)	10^-2
deci- (d)	10^-1
deca- (da)	10¹
hecto- (h)	10²
kilo- (k)	10³
mega- (M)	10⁶
giga- (G)	10⁹

What Units Should I Use?

We will introduce each unit we need as we go, along with some “touchstone values” to orient you such as those provided above. Again, we do expect you to memorize these values and you will be quizzed on them. These touchstone values are important to give you a sense of your results and to allow you to check if they are reasonable. For example, if you are solving for the energy in an atomic bond and get $5 \, \mathrm{J}$ , you are probably wrong as the tables above show that atomic bonds are in the $\rm eV$ range!

When working, you should, in general, use the SI units:

meters (m)
seconds (s)
kilograms (kg) ← NOTE it is kilograms NOT grams!
moles (mol)
Kelvin (K) for temperature ← We will explore this later!

as well as the units derived from these mks base units:

Newtons (N)
Joules (J)

In general, convert to these before doing any math and then, if needed, convert back. If you are really confident in what you are doing, it may be possible to solve problems in other units like Celsius $\rm ^\circ C$ or electron-Volts ( $\rm eV$ ), but be careful! The various constants in equations, such as $k_B$ often have units themselves and you need to use the value that matches the units you are working in.

Key Takeaways

When in doubt, convert to and work in the SI units and then convert back:
- meters (m)
- seconds (s)
- kilograms (kg)
- Kelvin (K)
- moles (mol)
- Newtons (N)
- Joules (J)
For each of these base units (particularly those that may be unfamiliar to those of us raised in the United States!) you will be expected to memorize some “touchstone” values to help orient you as you explore situations.
You are expected to have the SI prefixes from nano- to Giga- memorized and be able to convert between them.

Homework

Forces.
Units of pressure.
Mass and weight.
Touchstone amounts of energy.
Units of work and energy conversion.
SI unit prefixes.

https://en.wikipedia.org/wiki/Body_surface_area ↵
Flowers, P., Neth, E. J., Robinson, W. E., Theopold, K., & Langley, R. (2019). Measurements. In Atoms First 2e (p. 1.4). OpenStax. ↵

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Physics 131: What Is Physics? by Brokk Toggerson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

What is a Force?

Units of Force

Touchstone Amounts of Force

Another Useful Way to Think of Force: Pressure

Units of Pressure

The Atmosphere as a Unit of Pressure

Mass (As Opposed to the Force of Weight)

Touchstone Units of Mass (for those raised in the US!)

Work and Energy

Units of Work and Energy

Touchstone Amounts of Energy

A Note on Units

What Units Should I Use?

License

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